UNCLASSIFIED: Distribution Statement A: Approved for Public Release

2011 NDIA GROUND VEHICLE SYSTEMS ENGINEERING AND TECHNOLOGY

SYMPOSIUM

MODELING & SIMULATION, TESTING AND VALIDATION (MSTV) MINI-SYMPOSIUM AUGUST 9-11 DEARBORN, MICHIGAN

SAMPLING-BASED RBDO USING STOCHASTIC SENSITIVITY AND DYNAMIC KRIGING FOR BROADER ARMY APPLICATIONS

K.K. Choi, Ikjin Lee, Liang Zhao, and Yoojeong Noh

Department of Mechanical and Industrial Engineering The University of Iowa

Iowa City, IA

David Lamb and David Gorsich

US Army TARDEC Warren, MI

ABSTRACT

The University of Iowa has successfully developed Reliability-Based Design Optimization

(RBDO) method and software tools by utilizing the sensitivity analysis of the fatigue life; and

applied RBDO to Army ground vehicle components to obtain reliable optimum designs with significantly reduced weight and improved fatigue life. However, this method cannot be applied

to broader Army application problems due to lack of sensitivity analysis in many application

areas. Thus, for broader Army applications, a sampling-based RBDO method using surrogate

model has been developed recently. The Dynamic Kriging (DKG) method is used to generate

surrogate models, and a stochastic sensitivity analysis is used to compute the sensitivities of

probabilistic constraints with respect to independent and correlated random variables. Once the

DKG method accurately approximates the responses, there is no further approximation in the

estimation of the probabilistic constraints and stochastic sensitivities, and thus the sampling-

based RBDO can yield very accurate optimum design. For computational efficiency of the

sampling-based RBDO method for large-scale engineering problems, a parallel computing is

proposed. Numerical examples verify that the proposed sampling-based RBDO finds the optimum

designs very accurately and efficiently.

1. INTRODUCTION Significant advances in computing power provide design

engineers and decision-makers opportunities to explore

many more alternative simulation-based designs than they could do with hardware prototypes. However, when the

deterministic optimization method is used, the optimum

designs are pushed to the limits of the design constraint

boundaries, leaving very little or no room for physical input

uncertainties such as manufacturing dimensional

variabilities, material property variabilities, simulation

model uncertainties, and operational load variabilities, as

shown in Fig. 1. Thus, the deterministic optimum designs

obtained without consideration of these input uncertainties

are unreliable. Due to the extensive efforts of engineering

disciplines over last three decades, design guidelines and/or

standards have been modified to incorporate the concept of

uncertainty in the early design stage. Techniques have been

explored to incorporate uncertainty analysis at an affordable

computational cost and, more recently, to carry out design

optimization with the additional requirement of reliability,

which is referred to as reliability-based design optimization

(RBDO) [1-10].

For the most probable point (MPP) based RBDO, to alleviate the slow convergence of the reliability index

approach (RIA), the performance measure approach (PMA)

is developed by carrying out the inverse reliability analysis

[5-7] for a robust and efficient RBDO computational

process. For the inverse reliability analysis, the enhanced

hybrid mean value (HMV+) method has been developed

[11,12] to improve the computational efficiency and stability

for highly nonlinear and non-monotonic performance

functions [13,14]. The interpolation method of the HMV+

has been further improved by using the angle-based

parameter and has been integrated in the enriched

performance measure approach (PMA+) for RBDO [15,16].

The PMA+ method has been demonstrated to be very robust

and efficient [17].

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4. TITLE AND SUBTITLE Sampling-Based RBDO Uuing Stochastic Sensitivity andDynamic Kriging for Broader Army Applications

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6. AUTHOR(S) K.K. Choi; Ikjin Lee; Liang Zhao; Yoojeong Noh; David Lamb;David Gorsich

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7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) US Army RDECOM-TARDEC 6501 E 11 Mile Rd Warren, MI48397-5000, USA Department of Mechanical and IndustrialEngineering, The University of Iowa, Iowa City, IA, USa

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Proceedings of the 2011 Ground Vehicle Systems Engineering and Technology Symposium (GVSETS)

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The reliability analysis using FORM is inaccurate if the

performance function is highly nonlinear and multi-

dimensional. Although the reliability analysis using SORM

may be accurate, it is not easy to use since SORM requires

the second-order sensitivities. To overcome these

drawbacks by maintaining the efficiency of FORM and the

accuracy of SORM, the most probable point based

dimension reduction method (MPP-based DRM) has been

developed by the Iowa team [18-20]. For the input joint CDF model, the copula, which links

between the joint CDF and marginal CDFs, is used [21,22].

Since the copula requires only the marginal CDFs and

correlation parameters to generate the joint CDF, the joint

CDF can be obtained for practical industrial applications.

To identify the correct joint CDF type using the limited test

data, it is necessary to find a right copula that best describes

the paired sampled data. The two-step weight-based

Bayesian method, which selects a right copula among

candidate copulas based on the test data, is used [23].

Dynamics Analysis

FE Model

System Model

Dynamic Stress

Fatigue Life Contour RBDO/PBDO

DR

AW

Pro

/E

DA

DS

Dynamic Load AnalysisSolid Modeling

Component Stress Analysis

Durability Analysis and Reliability-Based Design Optimization

Terrain Model

Test Course MicroprofileATC Cross Country 3 Course - Left and Right Tracks

November, 1998

De-trended Data

-15

-10

-5

0

5

10

15

100 200 300 400 500 600 700

Distance - Feet

Dis

pla

cem

en

t -

In

ch

es

Right Track

Left Track

DR

AW

Refined Fatigue Analysis

AN

SYS

NA

STR

AN

DSO

RB

DO

/PB

DO

DP1

DP2

DP4

DP3

DP5

Fatigue

Material

Properties

Material

Property

Variabilities

Manufacturing

Dimensional

Variabilities

How to Optimize the Vehicle Design to Minimize/Reduce the Weight?

Under These Uncertainties, How to Achieve the Component Level Reliability?

Under These Uncertainties, How to Achieve the System Level Reliability?

Simulation Model

Uncertainties

Road Profile

(operational load)

Uncertainties

Figure 1. Ground Vehicle RBDO Process for Durability,

Reliability, and Weight Reduction

Over the years, the University of Iowa and U.S. Army

Tank Automotive Research, Development, and Engineering

Center (TARDEC) have been working together to develop a

simulation-based RBDO process to minimize the Army

ground vehicle weight while maintaining/improving

durability and system-level reliability requirements, as

shown in Fig. 1. The durability analysis process that

predicts fatigue failure of the ground vehicle components due to cyclic damage accumulations is a multidisciplinary

simulation process, requiring an integration of a CAD tool

and several CAE tools, such as multibody dynamic analysis,

FEA, and durability analysis; and a large amount of data

communication as shown in Fig. 1. In addition, the design

sensitivity analysis (DSA) [24,25] of the fatigue life with

respect to the design variables (random or deterministic) and

other random input variables is required to carry out the

inverse reliability analysis using HMV+ and design

optimization using PMA+ [14,15].

The propagation of the input uncertainty into the structural

fatigue life is complicate and thus a challenging task to

integrate the multidisciplinary software systems and develop

the RBDO process for durability optimization. The Iowa

research team has developed the DRAW software system

[26], which computes the transient dynamic stress and strain histories and fatigue life of mechanical components [27,28].

In addition, the DSO software system that computes the

design sensitivity of various performance measures,

including fatigue life, has been implemented by using the

continuum design sensitivity theories that the Iowa team has

developed [29-31]. These two software systems are

integrated with the commercial CAD-Pro/E [32], FEA-

NEiNastran [33], multibody dynamics code DADS [34], and

design optimization code DOT [35], along with the PMA+

RBDO software system that was developed by the Iowa-

TARDEC collaborative research team (see Fig. 1). This

integrated software system was successfully applied to

obtain reliable optimum designs with significantly reduced

weight and improved fatigue life of U.S. Army High

Mobility Trailer (HMT) drawbar [36], Stryker A-arm [37]

shown in Fig. 1, and HMMWV A-arm [38] components.

The system-level durability RBDO of the Army vehicle is a very compute-intensive process because it covers a number

of critical vehicle components; with each component

consisting of an FEA model possibly up-to hundreds of

thousands of DOF. Thus, it could take very long

computational time to carry out the vehicle system-level

durability RBDO on a single computer processor. The Iowa

and TARDEC research team initiated development and

testing of a parallelized DRAW-DSO-RBDO integrated

software system on the TARDEC High Performance Computing

(HPC) as shown in Fig. 2. The objective is to obtain a vehicle

system-level reliability-based optimum design for weight

reduction and durability of all critical components of the

Army ground vehicle in short computational time.

Successful scalability testing of the integrated and

parallelized software system was carried out on the

TARDEC HPC [39] to learn how to achieve the

computational speed-up. With the success of the MPP-based RBDO methods and

software tools, the Iowa team started extending them by

developing a sampling-based system level RBDO method

[40,41] to support broader ground vehicle applications as

shown in Fig. 3. For the sampling-based RBDO, the

stochastic sensitivity analysis [40] is developed to compute

sensitivities of probabilistic constraints with respect to

independent and correlated random design variables; and the

Dynamic Kriging (DKG) method is developed for surrogate

modeling [42].

Proceedings of the 2011 Ground Vehicle Systems Engineering and Technology Symposium (GVSETS)

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Optimizer (DOT)

Evaluation at Mean Value (Design) Point

DSO, DRAW

Probabilistic

Constraint EvaluationMPP search by RBDO

Evaluation of MPP

candidate

DSO, DRAW

Convergence

Check

STOP

……

……

Parallel Computing

on HPC

Identification of Active Constraints

Sh

are

d N

EiN

AS

TR

AN

Lic

enses

Probabilistic

Constraint EvaluationMPP search by RBDO

Probabilistic

Constraint EvaluationMPP search by RBDO

Evaluation of MPP

candidate

DSO, DRAW

Evaluation of MPP

candidate

DSO, DRAW

Figure 2. Parallelized DRAW-DSO-RBDO Integrated

Software System on TARDEC HPC

For Broader Applications of Modeling & Simulation-Based

RBDO, Each Discipline Needs to Develop Sensitivity Method

● Vehicle Weight Reduction and Durability

● Advanced & Hybrid Powertrain

● Electric Power & On-Board Electrification

● Robotic Systems

● Safety Analysis for Survivability & Design

● Human Centered Design

● Noise & Vibration

● Crashworthiness

● Manufacturing Processes

● MEMS & Microelectronics Packaging

● Nano & Micromechanics Based Materials

Modeling, …, Etc.

Figure 3. Future Goals: Broader Applications of

Reliability-Based Design Optimization

2. DYNAMIC KRIGING METHOD In the Kriging method, the outcomes are considered as a

realization of a stochastic process, and the predicted values

are derived by applying the stochastic process theory.

Consider n sample points with T nr

1 2[ , ,..., ] ,

n i X x x x x R , and n responses

T

1 2[ ( ), ( ),..., ( )]

ny y yy x x x with

1( )

iy x R . Thus, the

responses at samples are considered as a summation of two

parts as

y = Fβ + e

(1)

In Eq. (1), Fβ is the mean structure of the response, where

=[ ( )], 1,..., , 1,..., ,k i

f i n k K F x is an n×K design matrix;

and ( ) , 1,..., ,k

f k Kx represent the basis functions, which

are usually in a simple polynomial form, such as 2

1, , ,...,x x .

Also, T

1 2[ , ,..., ]

K β is the vector of regression

coefficients; and T

1 2[ ( ), ( ),..., ( )]

ne e ee x x x is a realization

of the stochastic process ( )e x that is assumed to have zero

mean [ ( )] 0i

E e x . The covariance structure is

2[ ( ) ( )] ( , , )

i j i jE e e Rx x θ x x , where

2 is the process

variance; 1 2, ,...,

nr θ is the correlation parameter

vector that has to be estimated by applying the maximum

likelihood estimator (MLE); and ( , , )i j

R θ x x is the

correlation function of the stochastic process [43]. Usually,

the correlation function is set to Gaussian form in

engineering applications and expressed as

2

, ,1

( , , ) exp( (x x ) )nr

i j l i l j ll

R

θ x x (2)

where ,

xi l

is the lth component of variable

ix . Under the

decomposition of Eq. (1) and θ, which is obtained to

maximize MLE,

12 T 1

2 22

12 exp ( ) ( )

2

n

L

R y Fβ R y Fβ

(3)

where R is the symmetric correlation matrix with i-jth

component ( , , ), , 1,..., ,ij i jR R i j n θ x x and

2 T 11( ) ( )

n

y Fβ R y Fβ and

1T 1 T 1

β F R F F R Y are

obtained from the generalized least square regression.

The objective of the Kriging method is to predict the

noise-free unbiased response at a new point of interest

denoted by x. This prediction of response at x is written as a

linear predictor as

T

krigy (x) w y (4)

whereT

1 2[ ( ), ( ),..., ( )]

nw w ww x x x denotes the n×1 weight

vector for prediction at x and is obtained using the unbiased

condition [ ( )] [ ( )]krigE y E yx x as [42]

1

2

1( )

2

w R r Fλ (5)

where λ is the Lagrange multiplier; and T

1[ ( , , ),..., ( , , )]

nR Rr θ x x θ x x is the correlation vector

between x and samples xi, i=1,…,n.

Under the assumption of the Gaussian process, the 1α level prediction interval of the response is obtained as

1 / 2 1 / 2

( ) ( ) ( ) ( ) ( )p pkrig krigy Z y y Z

x x x x x (6)

Proceedings of the 2011 Ground Vehicle Systems Engineering and Technology Symposium (GVSETS)

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where 1 /2

Z

is the 1α level quantile of the standard normal

distribution and 2( )

p x is the predicted variance at x given

by 2 2 T T 1 1 T 1( ) (1 ( ) )

p

x u F R F u r R r . Therefore, the

bandwidth of the prediction interval at a point of interest x0

is

1 / 2

( ) 2 ( )p

d Z

x x (7)

and this prediction interval will be used as an accuracy

measure to decide if the surrogate model is accurate or not.

Depending on the basis functions fk(x) used in Eq. (1), the

Kriging method is called the ordinary Kriging method, first-

order universal Kriging method, and second-order universal

Kriging method, where basis functions with constant terms

only, up to first-order polynomial terms, and up to second-

order polynomial terms, respectively, are used. For both the

ordinary and universal Kriging methods, the basis functions

do not change during the surrogate model generation

process. However, in general, the higher-order terms can

predict the nonlinear mean structure, and thus fixed-order

basis functions may not be good to describe the nonlinearity of the mean structure. On the other hand, it is also known

that, in some cases, the accuracy of the surrogate model may

deteriorate by using higher-order terms [44]. Therefore, it is

desirable to find the optimal set of polynomial basis

functions that could provide the most accurate surrogate

model.

First, the Dynamic Kriging (DKG) method dynamically

selects the optimal set of basis functions at each design point

so that the generated surrogate model has the best accuracy.

As explained above, the accuracy is measured using the

prediction interval bandwidth in Eq. (7). To apply the DKG

method to find the best basis function set, the highest-order

P must first be determined. With n samples given, Eq. (4)

can be solved when the total number of basis functions is

less than n, that is,

1Pnd P

n

. (8)

The highest order P of the polynomial that satisfies Eq. (8)

is determined. For example, if 10 samples are given (n=10)

for a 2-D problem, the highest-order P will be 3 to satisfy

Eq. (8) so that we can have basis functions up to third-order

polynomials as 1, x1, x2, x1x2, x12, x2

2, x1

2x2, x1x2

2, x1

3, and

x23. After deciding the highest-order P, the genetic

algorithm (GA) is used to find the best basis function set by

minimizing the Kriging process variance

2 T 11( ) ( )

n

y Fβ R y Fβ .

Second, a generalized pattern search algorithm is used to

find the optimal correlation parameter θ to maximize the

MLE in Eq. (3). Since it is not a gradient-based optimization method and guarantees the global convergence,

which is proven by Lewis and Torczon [45], the pattern

search algorithm is powerful enough to find the optimal θ.

Using the GA for the best basis function set and the pattern

search for the optimal θ, the studied examples show that the

DKG method outperforms existing surrogate model

generation methods including the universal Kriging method,

the polynomial response surface method, the radial basis

function method, the support vector regression method, and

the blind Kriging method [42].

3. SAMPLING-BASED RBDO The mathematical formulation of a general RBDO problem

is expressed as

Tar

L U

minimize Cost( )subject to [ ( ) 0] , 1, ,

, andjj F

nd nr

P G P j nc

d

X

d d d d R X R

(9)

where T{ } , 1~id i nd rvd μ(X ) is the design vector,

which is the mean value of the nd-dimensional random

variable vector T

1 2={ , , , }ndX X XrvX ; T={ , }rv rp

X X X

where rvX and rp

X stand for the random design variable

and random parameter components of the random input X ,

respectively; Tar

jFP is the target probability of failure for the

jth

constraint; and nc, nd, and nr are the number of

probabilistic constraints, design variables, and random

variables plus parameters, respectively.

A reliability analysis for both the component and system

levels involves calculation of the probability of failure,

denoted by PF, which is defined using a multi-dimensional

integral as

( ) [ ] ( ) ( ; )

( )

nr F

F

F FP P I f d

E I

XR

ψ X x x ψ x

X (10)

where ψ is a vector of distribution parameters, which

usually includes the mean (µ) and/or standard deviation (σ)

of the random input T

1, , nrX XX ; P represents a

probability measure; F is the failure set; ( ; )fX

x ψ is a

joint probability density function (PDF) of X; and E

represents the expectation operator. The failure set is

defined as : ( ) 0jF jG x x for component reliability

analysis of the jth

constraint function Gj(x), and

1: ( ) 0

nc

F jjG

x x and 1

: ( ) 0nc

F jjG

x x for

the series system and parallel system reliability analysis of

nc performance functions, respectively. ( )F

I x in Eq. (10)

is called an indicator function and defined as

1,

( )0,F

FI

otherwise

xx (11)

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In this paper, since the mean of X, T

1, , nd μ is used

as a design vector, the vector of distribution parameters ψ

is simply replaced with µ for the computation of the

probability of failure in Eq. (10).

Taking the partial derivative of probability of failure in

Eq. (10) with respect to the ith

design variable i yields

( )

( ) ( ; )nr F

F

i i

PI f d

XR

μx x μ x (12)

and the differential and integral operators can be

interchanged using the Leibniz‟s rule, giving

( ) ( ; )( )

ln ( ; )( ) ( ; )

ln ( ; )( )

nr F

nr F

F

F

i i

i

i

P fI d

fI f d

fE I

X

R

XX

R

X

μ x μx x

x μx x μ x

x μx

(13)

since ( )F

I x is not a function of i . The partial derivative

of the log function of the joint PDF in Eq. (13) with respect

to i is known as the first-order score function for i and is

denoted as

(1) ln ( ; )

( ; ) .i

i

fs

Xx μ

x μ (14)

To compute the probability of failure in Eq. (10) and the

sensitivity of probability of failure in Eq. (13), statistical

sampling such as the Monte Carlo simulation (MCS) at a

given design needs to be applied to true responses, which is computationally very expensive and almost prohibited.

Hence, instead of using the true responses, which are usually

obtained from computer simulations, the surrogate models

obtained using the DKG method are used.

Denote the surrogate model obtained by the DKG method

for the constraint function Gj(X) as ˆ ( )jG X . Then, by

carrying out the MCS using the conservative surrogate

model ˆ ( )jG X , the probabilistic constraints in Eq. (9) can be

approximated as

( ) Tar

ˆ

1

1[ ( ) 0] ( )

j jFj

Mm

F j F

m

P P G I PM

X x (15)

where M is the MCS sample size, ( )mx is the m

th realization

of X, and the failure set ˆjF for the surrogate model is

defined as ˆˆ : ( ) 0jF jG x x . Sensitivity of the

probabilistic constraint in Eq. (13) is obtained as

( ) (1) ( )

ˆ

1

1( ) ( ; )

j

iFj

MF m m

mi

PI s

M

x x μ (16)

where (1) ( )( ; )i

ms x μ is obtained using Eq. (14).

4. PARALLELIZATION OF SAMPLING-BASED RBDO

As explained in Section 2, the DKG method uses the

pattern search and genetic algorithm for more accurate

surrogate model generation. Hence, as the dimension of the

complex engineering system increases, the DKG method and

MCS for the reliability analysis using surrogate models

become computationally inefficient. Moreover, the number of samples required for computer simulations increases.

Therefore, a high performance computing strategy needs to

be implemented into the sampling-based RBDO procedure

to ensure it is applicable for large-scale complex engineering

applications.

For the sampling-based RBDO using the DKG method,

there exist three major places where the parallel computing

can be applicable: parallelization of surrogate model

generation for multiple constraints, parallelization of MCS

for multiple surrogate models, and parallelization of

computer simulations at samples. The Matlab parallel

computing toolbox and the parallel computing platform

(LSF-Platform) are utilized for the first two parallelizations

and for the last parallelization, respectively, of the sampling-

based RBDO with the DKG method.

Compared with the gradient-based or MPP-based RBDO,

the sampling-based RBDO has inherent advantages in terms of the parallelization. In the MPP-based RBDO, the

parallelization is limited to the number of constraints (nc),

which means that even if numerous client computers are

available it can only use nc client computers for the

parallelized computing. On the other hand, the sampling-

based RBDO can use as many client computers as the

number of samples, which is usually more than the number

of constraints for high dimensional problems. In addition,

the sampling-based RBDO has more places where the

parallelization can be applicable. Hence, in terms of the

parallel computing, the sampling-based RBDO is more

effective than the MPP-based RBDO [39].

4.1 Parallelization of Surrogate Models for Multiple Constraints in RBDO

A typical RBDO problem contains more than one

constraint. Since the surrogate model from the DKG method is computationally expensive for high dimensional large-

scale applications, it is desirable to carry out the surrogate

modeling for all constraints simultaneously, which leads to

the parallelization of surrogate modeling for multiple

constraints. This parallelization is conducted by using the

Matlab parallel computing tool-box.

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4.2 Parallelization of MCS in Reliability Analysis The MCS in the sampling-based RBDO is used to

calculate failure probabilities of performance functions as

well as their sensitivities with respect to design variables.

Usually, MCS requires a large number of samples for

accurate results. Moreover, since the prediction from the

DKG method at the MCS samples is implicit, a large

dimension matrix calculation is involved every time the

prediction is calculated at each MCS point. As the number of the MCS samples increases, the total computational time

for the reliability and sensitivity analysis increases as well.

Therefore, the parallelization of the MCS procedure is also

needed to reduce the large computational time. This

parallelization is also conducted by using the Matlab parallel

computing toolbox.

4.3 Parallelization of Computer Aided Engineering (CAE) at Samples

To generate surrogate models of the performance functions

in RBDO by the DKG method, it is required to evaluate the

performance functions at the sample points, which is usually

conducted by computer simulation. This procedure is

computationally intensive for large-scale complex

engineering applications. If the number of the computer

simulations is large for a large-scale engineering application,

apparently, the total computational time of conducting the computer simulations for all samples may become

unaffordable. Therefore, the parallelization is necessary for

this procedure. Usually the computer simulation is carried

out by general CAE commercial software and the

parallelization can be done by the parallel computing

platform (LSF-Platform).

4.4 Summary of Parallelization in Sampling-Based RBDO

With all the discussion above, we can obtain the entire

workflow of the parallelization in the sampling-based RBDO

shown in Fig. 4, where a “core” means a unit in a multiple-

core desktop and a “node” means one client computer in a

cluster network.

5. NUMERICAL EXAMPLES This section illustrates two design optimization examples –

a 2-D mathematical example and a 12-D M1A1 Abrams tank

roadarm − to see how the proposed sampling-based RBDO

works for an RBDO problem. The 2-D mathematical

example is used to show the accuracy and efficiency of the

proposed method since its analytic functions are available,

and thus the MCS is applicable for the comparison of the

probability of failure calculation. The 12-D M1A1 Abrams

tank roadarm is used to see how the proposed sampling-

based RBDO works for a high-dimensional engineering

application in terms of accuracy and efficiency. In addition,

using the roadarm example, the effectiveness of the

parallelization is also explained. For all examples, 1 million

testing points are used for the DKG method, and 500,000

MCS samples are used for the reliability and sensitivity

analysis and the MCS sample number increases to 1 million

when constraints are identified as active.

Figure 4. Workflow of Parallelization in RBDO

5.1 RBDO of 2-D Mathematical Problem Consider a 2-D mathematical RBDO problem, which is

formulated to 2 2

1 2 1 2

Tar

2 2

( 10) ( 10)minimize Cost( )

30 120subject to ( ( ( )) 0) 2.275%, 1 ~ 3

, andj

j F

L U

d d d d

P G P j

d

X d

d d d d R X R

(17)

where three constraint functions are expressed as 2

1 2

1

2 3 4

2

3 2

1 2

( ) 120

( ) 1 ( 6) ( 6) 0.6 ( 6)

80( ) 1

8 5

X XG

G Y Y Y Z

GX X

X

X

X

(1)

where1

2

0.9063 0.4226

0.4226 0.9063

XY

Z X

, and are drawn in Fig.

5. The properties of two random variables are shown in

Table 5, and they are correlated with the Clayton copula (τ=0.5). As shown in Eq. (17), the target probability of

failure (Tar

FP ) is 2.275% for all constraints.

As shown in Fig. 5 and Table 1, the initial design is d0

=

[5,5]T. At the initial design, the sampling-based

deterministic design optimization (DDO) is first used to find

the deterministic optimum, which is usually close to the

RBDO optimum, and the sampling-based RBDO is launched

at the deterministic optimum design. This approach is more

computationally efficient than launching the RBDO from the

beginning. As shown in Fig. 6, the DDO requires 30

Proceedings of the 2011 Ground Vehicle Systems Engineering and Technology Symposium (GVSETS)

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samples, which are marked as asterisks in the figure, for the

whole design iteration, and the deterministic optimum

design is exactly identical to the optimum design obtained

using analytic functions in Eq. (17). At the deterministic

optimum, the sampling-based RBDO is launched, with a

total of 18 samples are initially used for the first iteration of

the sampling-based RBDO. Twenty more samples, which

are marked as dots in Fig. 6, are generated for the sampling-

based RBDO whose result is shown in Table 2.

Figure 5. Shape of Constraint Functions

Table 1. Properties of Random Variables

Random

Variables Distribution d

L d

0 d

U

Standard

Deviation

X1 Normal 0.0 5.0 10.0 0.3

X2 Normal 0.0 5.0 10.0 0.3

Figure 6. Sample Profile for DDO and RBDO

Table 2 compares the numerical results of five different RBDO methods. The first three results are obtained from

the MPP-based RBDO, which requires sensitivities of

constraint functions for the MPP search and design

optimization. This MPP-based RBDO includes the FORM

[14] and the DRM [19] with three and five quadrature

points, which are denoted in Table 2 as DRM3 and DRM5,

respectively. The results of the last two rows are obtained

from the sampling-based RBDO, which uses the MCS for

the estimation of the probability of failure and its sensitivity.

The sampling-based RBDO using the DKG method is the

proposed method, and to compare the accuracy of the

proposed method, the result of the sampling-based RBDO using the analytic (true) functions given in Eq. (35) is also

shown in the table.

From the table, it can be seen that the probability of failure

of the second constraint (1.2835%) estimated by the MCS

with 50 million samples at the optimum design obtained

using the FORM is not close to the target probability of

failure (2.275%). This is because the second constraint is

highly nonlinear as shown in Fig. 6, and the FORM uses the

transformation from original X-space to standard normalized

U-space, which makes the constraint even more highly

nonlinear due to the correlated nonlinear input. For highly

nonlinear functions, the FORM cannot accurately estimate

the probability of failure since it uses the linear

approximation of the nonlinear functions at the MPP in U-

space. To improve the accuracy of the probability of failure

at the optimum design, the MPP-based DRM with three or

five quadrature points can be used [19]; Table 2 shows that the MPP-based DRM indeed improves the accuracy of the

probability of failure at the optimum design with more

function evaluations. However, to obtain a more accurate

optimum design, more quadrature points are required, such

as the DRM7, etc. To obtain the optimum design, the

FORM uses 52 function evaluations and 52×2=104

sensitivity calculations, whereas the MPP-based DRM with

five quadrature points uses 146 function evaluations and

102×2=104 sensitivity calculations, and the number of

function evaluations for the MPP-based DRM will be

increased as the number of quadrature points increases [19].

Table 2. Comparison of Various RBDOs ( = 2.275%Tar

FP )

On the other hand, the sampling-based RBDO shows very

accurate optimum design since the optimum design is very

Methods Cost Optimum

Design

MCS (50M) Number of Function

Evaluations PF1, % PF2, %

MPP-Based

RBDO

FORM -1.8742 5.0026, 1.6165 2.3022 1.2835 52+52×2

DRM3 -1.8794 5.0315, 1.6050 2.2912 1.7496 128+106×2

DRM5 -1.8821 5.0454, 1.5988 2.2621 2.0183 146+102×2

Sampling-

Based RBDO

DKG -1.8845 5.0576, 1.5936 2.2716 2.2869 50

Analytic

Function -1.8853 5.0541, 1.5918 2.2912 2.2791 N.A.

Proceedings of the 2011 Ground Vehicle Systems Engineering and Technology Symposium (GVSETS)

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close to the optimum design obtained using the analytic

functions. However, it requires only 50 samples, which is

even less than the FORM, for the accurate optimum design

without requiring the sensitivity of the performance

functions. The sampling-based RBDO can obtain a very

accurate optimum design because it does not use any

approximation on the calculation of the probability of

failure, unlike the FORM and MPP-based DRM, and the

DKG method generates very accurate surrogate models. In addition, it can be said that the proposed efficiency strategies

indeed work in this example. Therefore, once surrogate

models for constraint functions are accurate enough, the

proposed sampling-based RBDO could obtain a very

accurate optimum design with good efficiency.

5.2 RBDO of M1A1 Abrams Tank Roadarm The roadarm of the M1A1 Abrams tank [19] is used to

compare two approaches: the MPP-based RBDO, which

requires sensitivities of performance functions, and the

sampling-based RBDO, which does not require sensitivities

of performance functions, for the component RBDO. The

roadarm is modeled using 1572 eight-node isoparametric

finite elements (SOLID45) and four beam elements

(BEAM44) of ANSYS [46], as shown in Fig. 7, and is made

of S4340 steel with Young‟s modulus E=3.0×107 psi and

Poisson‟s ratio ν=0.3. The durability analysis of the roadarm is carried out using Durability and Reliability Analysis

Workspace (DRAW) [26] to obtain the fatigue life contour.

The fatigue lives at the critical nodes are shown in Fig. 8,

which are chosen as the design constraints of the RBDO.

Figure 7. Finite Element Model of Roadarm

Figure 8. Fatigue Life Contour at Critical Nodes of

Roadarm

The shape design variables are shown in Fig. 9. Eight

shape design variables characterize four cross-sectional

shapes of the roadarm. The widths (x1-direction) of the

cross-sectional shapes are defined by the design variables d1,

d3, d5, and d7 at intersections 1, 2, 3, and 4, respectively, and

the heights (x3-direction) of the cross-sectional shapes are

defined using the remaining four design variables. Eight

shape design variables are listed in Table 3 and are assumed to be independent random variables.

Figure 9. Shape Design Variables for Roadarm

For the input fatigue material properties, since the

statistical information on S4340 steel other than its nominal

value is not available, it is necessary to assume the statistical

information on S4340 steel. The strain-life relationship is

given by the classical Coffin-Manson equation as [47]

(2 ) (2 )2 2 2

s dp f b ce

f f fN N

E

(19)

where f

and bs are the fatigue strength coefficient and

exponent, respectively; f

and cd are the fatigue ductility

coefficient and exponent, respectively; Nf is the fatigue

initiation life; and E is the Young‟s modulus. It is known

that f

and f

follow the lognormal distribution and bs and

cd follow the normal distribution. Furthermore, it is also

known that f

, bs, and f

, cd are highly negatively

correlated [22]. For the correlated fatigue material

properties, it is assumed that f

and bs follow the Gaussian

copula with ρ=−0.828 and that f

and cd follow the Frank

copula with τ=−0.906 [22]. For the standard deviations of

Proceedings of the 2011 Ground Vehicle Systems Engineering and Technology Symposium (GVSETS)

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S4340 steel, 3% coefficients of variation (COV) for fatigue

material properties are assumed as shown in Table 3. It is

known that fatigue strength parameters and fatigue ductility

parameters are not correlated each other.

Table 3. Random Variables and Fatigue Material

Properties

Random

Variables

Lower

Bound

dL

Initial

Design

d0

Upper

Bound

dU COV

Distribution

Type

d1, in. 1.350 1.750 2.150

1% Normal

d2, in. 2.650 3.250 3.750

d3, in. 1.350 1.750 2.150

d4, in. 2.570 3.170 3.670 d5, in. 1.356 1.756 2.156

d6, in. 2.438 3.038 3.538

d7, in. 1.352 1.752 2.152

d8, in. 2.508 2.908 3.408

Fatigue Material Properties

Non-design Uncertainties Mean COV Distribution

Type

Fatigue Strength Coefficient,

f, psi 177000

3%

Lognormal

Fatigue Strength Exponent, bs -0.0730 Normal

Fatigue Ductility Coefficient, f 0.4100 Lognormal

Fatigue Ductility Exponent, cd -0.6000 Normal

The RBDO for the M1A1 Abrams tank roadarm is formulated to

Tar

L U 8 12

minimize Cost( )

subject to [ ( ) 0] , 1, ,13

, andj

j FP G P j

d

X

d d d d R X R

(20)

where

Tar

Cost( ) : Weight of Roadarm

( )( ) 1 , 1 ~ 13

( ) : Crack Initiation Fatigue Life,

: Crack Initiation Target Fatigue Life (=5 years)

2.275%

j

t

t

F

LG j

L

L

L

P

d

dd

d (21)

For the sampling-based DDO, 15 samples are used as the

initial number of samples in the local window. Smaller local

window is used for the DDO since the accuracy of the

surrogate model near a given design is required for

sensitivity calculation and there is no need of reliability

analysis. After 11 iterations, the sampling-based DDO

converged to the optimum design, using 135 samples. The

optimum design obtained using the sampling-based DDO is

almost identical with the optimum design obtained using the

sensitivity-based DDO as shown in Table 4. The sensitivity-

based DDO requires 11 function and 11 sensitivity

evaluations as shown in Table 4, where F.E. stands for „function evaluation‟. One sensitivity evaluation includes

sensitivity calculations for all design variables, so it requires

11×8=88 sensitivity calculations in this example, whereas

the sampling-based DDO requires a total of 135 samples for

the surrogate model generation using the DKG method.

Table 4. Comparison of Optimum Designs

Design Variable

Initial Sensitivity-Based Sampling-Based

DDO RBDO DDO RBDO

d1 1.750 1.653 1.711 1.653 1.705

d2 3.250 2.650 2.650 2.650 2.650

d3 1.750 1.922 1.943 1.922 1.941 d4 3.170 2.570 2.570 2.570 2.570

d5 1.756 1.478 1.514 1.478 1.508

d6 3.038 3.287 3.348 3.287 3.352

d7 1.752 1.630 1.691 1.630 1.702 d8 2.908 2.508 2.508 2.508 2.508

# of F.E. 11+11×8 85+85×12 135 691

Active Constraints

1,3,5,8,12 1,3,5,8,12 1,3,5,8,12 1,3,5,8,12

Cost 515.09 466.80 474.20 466.81 474.60

The sampling-based RBDO is launched at the DDO. In

this case, samples used for the DDO cannot be used for the

RBDO unlike the mathematical example because the

dimension of the DDO is 8, whereas the dimension of the

RBDO is 12. The number of initial samples within the local

window is 200. It is found that 4 out of 13 performance

functions are very feasible at the deterministic optimum.

Hence, surrogate models for those performance functions are

not generated to save the computation time. Table 4 also

compares two RBDO optimum designs obtained using the

sensitivity-based and sampling-based RBDO. The FORM is

used for the sensitivity-based RBDO.

From the table, it can be seen that two optimum designs

are very close to each other. At the optimum design

obtained using the sampling-based RBDO, the MSE of the surrogate model is 0.0062, which is much less than the target

MSE (0.01) and means that surrogate model at the optimum

design is accurate enough. To obtain the optimum design

using the FORM, 85 function and 85×12=1020 sensitivity

evaluations are used since there exist 8 random design

variables and 4 random parameters, whereas the sampling-

based RBDO uses 691 samples, which means 691 function

evaluations, to find the optimum design.

5.3 Efficiency of Parallel Computing It is noted that one license of the Matlab parallel

computing toolbox allows 8 cores working simultaneously,

therefore 8 surrogate models for constraints are generated at

the same time. Thus, using the parallelization explained in

Sections 4.1 and 4.2, the computation time is maximally 8

times faster than the one without the parallelization.

However, the number of cores used for the parallelization

Proceedings of the 2011 Ground Vehicle Systems Engineering and Technology Symposium (GVSETS)

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can be extended if parallel computing software other than

the Matlab toolbox is utilized.

To demonstrate the improvement of the efficiency by

applying the parallelization to the sampling-based RBDO

using the DKG method, the same M1A1 Abrams tank

roadarm example in Section 5.2 is used. The comparison

between the parallel computing and the original serial

computing is carried out by running the reliability analysis at

the deterministic optimum design with 250 points in the local window. Nine constraints are identified as active or

violated constraints at the deterministic optimum design and

thus surrogate models are generated for those 9 constraints

only.

As shown in Table 5, the parallel computing reduces the

computational time by 72.7% for surrogate modeling and

67.4% for the MCS. The reason that the reduction of the

computational time for the surrogate modeling and MCS is

not exact 8 times compared with the serial one is that there

are nine active or violated constraints used to generate the

surrogate models while eight cores are used for the

parallelization. Therefore, it includes 2 iterations for the

parallel computing, 1st iteration for the 1

st ~ 8

th constraints

and 2nd

iteration for the 9th constraint only.

Table 5. Comparison of Computational Time

Method No. of

Samples

No. of

Constraints

Surrogate

Modeling,

sec.

MCS,

sec.

Parallel 250 9

129.73 337.82

Serial 474.43 1034.79

In this example, the 3rd

parallelization explained in Section

4.3, which is the parallelization of FEA, is not used; instead,

only one client computer is used for the test. If 50 client

computers are used for this test, computer simulation time

for the FEA will be almost 50 times faster since the data

transmission for the parallelization is negligible.

Furthermore, the MPP-based RBDO can use maximally 13

client computers only since there are 13 constraints for the

M1A1 Abrams tank roadarm. On the other hand, the

sampling-based RBDO can use as many client computers as

the number of samples used. The parallelization of FEA is being tested on the U.S. Army TARDEC high performance

computing (HPC) system.

6. CONCLUSION For broader applications, sampling-based RBDO using the

DKG method for surrogate model generation and the score

function for probability of failure and its sensitivity analysis

is proposed in this study. The proposed sampling-based

RBDO does not use any approximation on the calculation of

the probability of failure and its sensitivity except for

statistical noise due to the MCS, which can be easily solved

by increasing the MCS sample set. Furthermore, the

proposed method does not use the transformation from the

original X-space to the standard normal U-space, which

makes performance functions become more highly

nonlinear, especially when random inputs are correlated.

Therefore, the proposed sampling-based RBDO is more

accurate than the sensitivity-based RBDO, which uses

approximation and transformation for the probability of failure estimation once surrogate models are sufficiently

accurate. The accuracy issue of surrogate models is resolved

in this paper by the use of the DKG method. In addition, to

enhance the efficiency of the proposed method for high-

dimensional problems, the parallel computing is used. Even

though only component level RBDO examples are treated in

this paper, the proposed sampling-based RBDO can be

easily extended to the system-level RBDO by using the

failure set of either series or parallel or mixed system.

Numerical examples are illustrated to demonstrate how the

proposed sampling-based RBDO works compared with the

sensitivity-based RBDO. The 2-D mathematical example

shows that the proposed method is more accurate and even

more efficient than the sensitivity-based RBDO, which

means the proposed method is very powerful when the

dimension of problems is low. For high-dimensional

problems such as the M1A1 Abrams tank roadarm used in the paper, the sampling-based RBDO still yields an accurate

optimum design. However, it may require more function

evaluation, which can be resolved by parallelizing the

computation procedure.

7. ACKNOWLEDGEMENT This research was primarily supported by the U.S. Army

Research Office (Project W911NF-09-1-0250). Some

funding for this work was provided by the Automotive

Research Center, a U.S. Army Center of Excellence for

Modeling and Simulation of Ground Vehicles led by the

University of Michigan.

Disclaimer: Reference herein to any specific commercial company, product, process, or service by trade name,

trademark, manufacturer, or otherwise, does not necessarily

constitute or imply its endorsement, recommendation, or

favoring by the United States Government or the

Department of the Army (DoA). The opinions of the authors

expressed herein do not necessarily state or reflect those of the United States Government or the DoA, and shall not be

used for advertising or product endorsement purposes.

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